Problem 17 - Elementary Algebra

Question

Set up an equation and solve each of the following problems. (Objectives 1-3) Suppose that we want to mix some \(30 \%\) alcohol solution with some \(50 \%\) alcohol solution to obtain 10 quarts of a \(35 \%\) solution. How many quarts of each kind should we use?

Step-by-Step Solution

Answer

Use 7.5 quarts of 30% solution and 2.5 quarts of 50% solution.

Step 1: Understanding the Problem

We need to determine how many quarts of 30% alcohol solution and 50% alcohol solution are required to make 10 quarts of a 35% alcohol solution.

Step 2: Setting Up Variables

Let \( x \) be the number of quarts of the 30% alcohol solution. Then, the number of quarts of the 50% alcohol solution will be \( 10 - x \) since the total must be 10 quarts.

Step 3: Writing the Equation

The total amount of alcohol from both solutions should equal the amount of alcohol in 10 quarts of 35% solution. The equation becomes:\[ 0.30x + 0.50(10 - x) = 0.35 \times 10 \]

Step 4: Simplifying the Equation

Expand and simplify the equation:\[ 0.30x + 5 - 0.50x = 3.5 \] Combine like terms:\[ -0.20x + 5 = 3.5 \]

Step 5: Solving for x

Isolate \( x \) by first subtracting 5 from both sides: \[ -0.20x = 3.5 - 5 \] \[ -0.20x = -1.5 \] Then divide by -0.20: \[ x = \frac{-1.5}{-0.20} = 7.5 \].

Step 6: Finding the Second Quantity

Since \( x = 7.5 \), there are 7.5 quarts of the 30% solution. Then, the amount of the 50% solution is \( 10 - 7.5 = 2.5 \) quarts.

Key Concepts

Linear Equations

Linear equations form the backbone of many algebra word problems; by understanding how to set up and solve these equations, we can find solutions to complex problems easily. A linear equation is typically in the form:

• \( ax + b = c \)

where \( a \) and \( b \) are constants, \( x \) is the variable, and \( c \) is the result. In mixing problems, the equation represents the total amount of each type you're mixing reaching a certain concentration.
To solve for the variable, you apply operations like addition, subtraction, division, and multiplication, while working step by step to isolate \( x \). This systematic method allows you to balance both sides and find the value of the unknown quantity, telling you exactly what is needed to reach the desired outcome.

Solution Mixtures

Solution mixtures refer to combining substances with different concentrations to achieve a specific mixture. Here, we mix two solutions with different alcohol concentrations to achieve a desired final concentration.
When setting up these types of problems, each solution's contribution is considered based on its concentration and the amount used.

• Let \( x \) be the amount of one solution.
• Then find the complement for the second solution using the total volume desired, for example, \( 10 - x \).
• Add the effects of each, proportionate to their concentration, to create an equation.

The goal is to achieve the exact concentration specified for the final mixture through proper measurement and calculation. Understanding the relationship between each part of the mixture and how they add up greatly helps in solving these problems.

Percent Concentration

Percent concentration is a key concept in solving mixture problems. It measures the amount of a substance present in a solution relative to the entire solution's volume, usually in percentage form.
To calculate the concentration, consider:

• The amount of substance (like alcohol) in the solution, expressed as a percentage.
• The total volume of the solution.
• For our problem, the percent concentrations are used to determine how much alcohol each mixture contributes to the final solution.

By multiplying the volume of each solution by its concentration percentage and equating the sum to the desired mix concentration, one can accurately determine the needed volumes of each solution. This careful calculation ensures that the solution meets the specified criteria, making percent concentration a vital part of mixture problems.